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6.3 The Hayward energy

We saw that EH can be non-zero even in the Minkowski spacetime. This may motivate considering the following expression
V~ ---------( gf ) Area(S)- -1- ' ' ' I (S) := 16pG2 1 + 4p (2rr - ss - ss ) dS = V~ --------- gf S Area(S)--1- ( ) = 16pG2 4p - y2 - y2'+ 2f11 + 2/\ dS. S
(Thus the integrand is 14(F + F ), where F is given by Equation (23View Equation).) By the Gauss equation this is zero in flat spacetime, furthermore, it is not difficult to see that its limit at the spatial infinity is still the ADM energy. However, using the second expression of I(S), one can see that its limit at the future null infinity is the Newman-Unti rather than the Bondi-Sachs energy.

In the literature there is another modification of the Hawking energy, due to Hayward [183Jump To The Next Citation Point]. His suggestion is essentially I(S) with the only difference that the integrands above contain an additional term, namely the square of the anholonomicity - wawa (see Sections 4.1.8 and 11.2.1). However, we saw that w a is a boost gauge dependent quantity, thus the physical significance of this suggestion is questionable unless a natural boost gauge choice, e.g. in the form of a preferred foliation, is made. (Such a boost gauge might be that given by the main extrinsic curvature vector Qa and Q~a discussed in Section 4.1.2.) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [63Jump To The Next Citation Point65] seems to be gauge invariant, this is due only to an implicit gauge choice. The correct, general GHP form of the extra term is a ' ' - waw = 2(b - b )(b - b ). If, however, the GHP spinor dyad is fixed as in the large sphere or in the small sphere calculations, then b - b'= t = - t', and hence the extra term is, in fact, 2t t.

Taking into account that t = O(r -2) near the future null infinity (see for example [338Jump To The Next Citation Point]), it is immediate from the remark on the asymptotic behaviour of I(S) above that the Hayward energy tends to the Newman-Unti instead of the Bondi-Sachs energy at the future null infinity. The Hayward energy has been calculated for small spheres both in non-vacuum and vacuum [63]. In non-vacuum it gives the expected value 4p 3 a b 3 r Tabt t. However, in vacuum it is -8- 5 ab cd - 45Gr Tabcdtt t t, which is negative.


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